The Affine Cipher is a type of monoalphabetic substitution cipher that utilizes mathematical operations to encrypt and decrypt messages. The encryption process in the Affine Cipher involves two keys, typically denoted as and
, and follows the formula:
where represents the encrypted letter,
is the numerical equivalent of the plaintext letter (with
),
and
are keys, and 26 is the modulus, corresponding to the number of letters in the English alphabet.
For decryption, the formula used is:
where represents the decrypted letter,
is the numerical equivalent of the ciphertext letter, and
is the modular multiplicative inverse of
modulo 26.
The necessity for the key to be coprime with the modulus 26 stems from the requirement to find the modular multiplicative inverse of
during the decryption process. A number
is said to be coprime (or relatively prime) with another number
if their greatest common divisor (GCD) is 1, i.e.,
.
Why Key
Must Be Coprime with 26
1. Existence of Modular Inverse:
The modular multiplicative inverse of modulo 26 exists if and only if
and 26 are coprime. The modular inverse
is a number such that:
If is not coprime with 26, the equation above has no solution, meaning
does not exist. Without the inverse, the decryption formula cannot be applied, rendering the decryption process infeasible.
2. Ensuring Unique Decryption:
When and 26 are coprime, each ciphertext letter maps uniquely to one plaintext letter. If
is not coprime with 26, the mapping between plaintext and ciphertext is not bijective, leading to multiple plaintext letters potentially mapping to the same ciphertext letter. This ambiguity undermines the cipher's reliability and security.
3. Mathematical Properties:
The coprimality condition ensures that the linear transformation applied by the Affine Cipher is invertible. The mathematical foundation of the Affine Cipher relies on linear algebra principles, where invertibility is crucial for reversing the transformation.
Implications if
is not Coprime with 26
1. Non-existence of Inverse:
If is not coprime with 26, the modular inverse
does not exist. For example, if
, which shares a common factor of 13 with 26, there is no integer
such that:
Consequently, it is impossible to decrypt the ciphertext using the standard decryption formula.
2. Ambiguity in Decryption:
Suppose , which is not coprime with 26 (since
). In this case, the mapping is not one-to-one. For instance, both plaintext letters 'a' (
) and 'n' (
) would map to the same ciphertext letter:
This results in a ciphertext where multiple plaintext letters correspond to the same ciphertext letter, creating ambiguity and making it impossible to uniquely determine the original plaintext.
3. Security Weakness:
The security of the Affine Cipher relies on the difficulty of deducing the keys from the ciphertext. If is not coprime with 26, the cipher becomes more predictable and easier to break. The redundancy introduced by non-coprime
values reduces the effective key space, making the cipher more vulnerable to cryptanalysis.
Example of Valid and Invalid Keys
– Valid Key Example:
Let and
.
Since ,
is coprime with 26. The encryption function is:
To decrypt, we need the modular inverse of 5 modulo 26. Using the Extended Euclidean Algorithm, we find that . The decryption function is:
This ensures that each ciphertext letter can be uniquely decrypted to its corresponding plaintext letter.
– Invalid Key Example:
Let and
.
Since ,
is not coprime with 26. The encryption function is:
Attempting to decrypt, we find that 6 does not have a modular inverse modulo 26. Therefore, the decryption process cannot be performed using the standard formula, leading to an unusable cipher.
Conclusion
The requirement for the key in the Affine Cipher to be coprime with the modulus 26 is fundamental to the cipher's functionality. It ensures the existence of a modular inverse, which is essential for the decryption process. Without this condition, the cipher fails to provide a unique and reversible mapping between plaintext and ciphertext, compromising both the usability and security of the encryption scheme.
Other recent questions and answers regarding EITC/IS/CCF Classical Cryptography Fundamentals:
- In the context of public-key cryptography, how do the roles of the public key and private key differ in the RSA cryptosystem, and why is it important that the private key remains confidential?
- Why is the security of the RSA cryptosystem dependent on the difficulty of factoring large composite numbers, and how does this influence the recommended key sizes?
- How does the method of "Exponentiation by Squaring" optimize the process of modular exponentiation in RSA, and what are the key steps of this algorithm?
- What are the steps involved in the key generation process of the RSA cryptosystem, and why is the selection of large prime numbers crucial?
- How does the RSA cryptosystem address the problem of secure key distribution that is inherent in symmetric cryptographic systems?
- How does the calculation of the modular inverse using the Extended Euclidean Algorithm facilitate secure communication in public-key cryptography? Provide a step-by-step example to illustrate the process.
- What is the Extended Euclidean Algorithm, and how does it differ from the standard Euclidean Algorithm? Explain its significance in finding modular inverses in cryptographic applications.
- How does Euler's Theorem relate to the RSA encryption algorithm, and why is it fundamental to the security of RSA?
- What is Euler's Phi Function, and how is it calculated for a given integer ( n )? Give examples for both a prime number and a product of two distinct primes.
- How does the Euclidean Algorithm work to find the greatest common divisor (GCD) of two integers, and why is it important in cryptographic protocols?
View more questions and answers in EITC/IS/CCF Classical Cryptography Fundamentals